Related papers: Multivariable automatic arrays and transcendence

We show that if a Laurent series $f\in\mathbb{C}((t))$ satisfies a particular kind of linear iterative equation, then $f$ is either a rational function or it is differentially transcendental over $\mathbb{C}(t)$. This condition is more…

In this paper possible completion $^*R_{d}$ of the Robinson non-archimedean field $^*R$ constructed by Dedekind sections. Given an class of analytic functions of one complex variable $f \in C[z]$,we investigate the arithmetic nature of the…

Reversible weighted automata are introduced and considered in a specific setting where the weights are taken from a nontrivial locally finite commutative ring such as a finite field. It is shown that the supports of series realised by such…

A rationally dynamically algebraic (RDA) power series is one that arises as (a component of) the solution of a system of differential equations of the form $\boldsymbol{y}' = F(\boldsymbol{y})$, where $F$ is a vector of rational functions…

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed…

We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental…

A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers…

Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of $p$-adic integers (or integers) is $p$-automatic when reduced modulo $p^\alpha$. Previously, the best known bound on the minimal automaton size for such a…

We consider a space of complex polynomials of degree $n\ge 3$ with $n-1$ distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent…

In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles or related to more general basis with multiple weights…

A (noncommutative) P\'olya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^\times$. We show that rational P\'olya series are unambiguous rational series,…

Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions.…

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…

We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…

About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering…

Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\cos (\pi r_1)$, and $\cos (\pi r_2)$ to be rationally independent. Extending a classical fact sometimes…

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…

In this paper we deal with three models of weighted automata that take weights in the field of real numbers. The first of these models are classical weighted finite automata, the second one are crisp-deterministic weighted automata, and the…

In this paper we study the existence of higher dimensional arithmetic progression in Meyer sets. We show that the case when the ratios are linearly dependent over $\ZZ$ is trivial, and focus on arithmetic progressions for which the ratios…